Why used $\cos\theta$ for $\text{y}$ axis or, gravitational force?

In summary, the conversation discusses a scenario where two masses, M1 and M2, are connected by a string and a pulley. M1 is held on an inclined plane with angle θ, while M2 hangs over the side. The coefficient of kinetic friction between M1 and the plane is µ. When M1 is released from rest, the masses experience different forces, including gravitational force and tension in the string. The tension and acceleration of the masses are calculated using equations of motion. The conversation also mentions the use of cosine for gravitational force, which is due to the concept of vector components.
  • #1
Istiak
158
12
Homework Statement
Mass M1 is held on a plane with inclination
angle θ, and mass M2 hangs over the side. The two masses are connected by a
massless string which runs over a massless pulley (see Fig. 3.1). The coefficient of
kinetic friction between M1 and the plane is µ. M1 is released from rest. Assuming that
M2 is sufficiently large so that M1 gets pulled up the plane, what is the acceleration
of the masses? What is the tension in the string?
Relevant Equations
F=ma
>![figure 3.2](https://physics.codidact.com/uploads/B5XdWq6GbB4vwyADQdALaCrC)![figure 3.1](https://physics.codidact.com/uploads/pkmWFgoesvQaiAfv5yKj6ynB)<br/>
>Mass M1 is held on a plane with inclination
angle θ, and mass M2 hangs over the side. The two masses are connected by a
massless string which runs over a massless pulley (see Fig. 3.1). The coefficient of
kinetic friction between M1 and the plane is µ. M1 is released from rest. Assuming that
M2 is sufficiently large so that M1 gets pulled up the plane, what is the acceleration
of the masses? What is the tension in the string?

Then, they were writing force of that figure.

$$T-f-M_1g\sin \theta = M_1a$$
$$N-M_1g\cos \theta=0$$
$$M_2g-T=M_2a$$

In the second equation they wrote that $$M_1g\cos \theta$$

Usually, $\cos$ is used when we think of $\text{x}$ axis. Since, $$\cos \theta=\frac{\color{blue}\text{base}}{\text{hypotenuse}}$$
But, gravitational force is forever through $\text{y}$ axis. Although, why they used $\cos\theta$ for gravitational force.
 
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  • #2
Force is a vector and can be decomposed into components tangential to and normal to a slope. That's why an object tends to move down a slope with less than the free fall acceleration.

As in your other post, it looks like it's the concept of vector components you are missing.
 
  • #3
Istiakshovon said:
Usually, ##\cos## is used when we think of ##{x}## axis.
That is the case when starting with something (a force, a displacement..) at angle theta to the horizontal and finding the horizontal component: ##x=r\cos(\theta)##.
In this case, we are starting something vertical (weight) and finding its component normal to the slope. That is the same as the angle between the plane and the horizontal.
 
  • Like
Likes Istiak

FAQ: Why used $\cos\theta$ for $\text{y}$ axis or, gravitational force?

Why is cosine used for the y-axis?

The cosine function is used for the y-axis because it represents the vertical component of a vector. In physics, the y-axis is typically used to represent the vertical direction, and the cosine function is a mathematical tool that helps us determine the vertical component of a vector.

Why is cosine used for gravitational force?

The cosine function is used for gravitational force because it helps us determine the vertical component of the force. In physics, gravitational force is often represented by a vector, and the cosine function allows us to calculate the vertical component of that vector, which is important for understanding the overall force.

Can other trigonometric functions be used for the y-axis or gravitational force?

Yes, other trigonometric functions such as sine and tangent can also be used for the y-axis or gravitational force. However, cosine is often preferred because it is specifically used to calculate the vertical component of a vector, making it a more natural choice for these applications.

How does using cosine for the y-axis or gravitational force impact calculations?

Using cosine for the y-axis or gravitational force does not significantly impact calculations. It simply allows us to determine the vertical component of a vector or force, which is necessary for understanding the overall magnitude and direction of the vector or force.

Are there any other applications of cosine in physics?

Yes, cosine has many other applications in physics. It is commonly used in calculating the trajectory of projectiles, determining the angle of refraction in optics, and analyzing the motion of pendulums, among other things.

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